How to wrap a complex object with one continuous image?
What you're describing is UV Mapping. You need to create seams and unwrap a map of the 3D object. That doesn't mean the seams need to be visible. You can texture paint right over the seams and they won't show.
Image files are 2D, and this is why we must do this.

In the following screen shot, notice how I have painted green over a seam in Texture Paint Mode and on the 3D model it appears to be seamless? The break in continuity occurs on the 2D image instead, which you can see below.



Here's a .blend file of the screen shot if you want to examine it further.



You might be thinking "that's fine if you're painting, but what about photo textures?" There's a solution for that too. You can use "Stencil" Brush Mapping to paint an image onto your mesh (onto its texture, really), and also use the Clone Brush to blend the image across the seams. 

Here is an example of how stenciling and cloning can be used to paint photo textures onto your 3D objects:



This is the result of just a couple of minutes spent playing around, so you can imagine the possibilities.